Fast Bayesian Estimation for the Four-Parameter Logistic Model (4PLM)

Abstract

There is a rekindled interest in the four-parameter logistic item response model (4PLM) after three decades of neglect among the psychometrics community. Recent breakthroughs in item calibration include the Gibbs sampler specially made for 4PLM and the Bayes modal estimation (BME) method as implemented in the R package mirt. Unfortunately, the MCMC is often time-consuming, while the BME method suffers from instability due to the prior settings. This paper proposes an alternative BME method, the Bayesian Expectation-Maximization-Maximization-Maximization (BE3M) method, which is developed from by combining an augmented variable formulation of the 4PLM and a mixture model conceptualization of the 3PLM. The simulation shows that the BE3M can produce estimates as accurately as the Gibbs sampling method and as fast as the EM algorithm. A real data example is also provided.

Mixture‐modelling‐based Bayesian MH‐RM algorithm for the multidimensional 4PLM

“…This study can be regarded as a natural extension of previous research in a multidimensional case. On the one hand, the MM-MH-RM algorithm for the M4PLM inherits the advantages of the mixture-modelling approach (Zheng et al, 2021) and can yield robust estimates with guaranteed convergence rates. On the other hand, using a stochastic approximation, the MM-MH-RM algorithm can handle high-dimensional data at much lower computation times than the MCMC methods.…”

“…Culpepper (2015) and Zheng et al (2021) analysed the second dimension (last 10 items) via unidimensional 4PLM and demonstrated that unslipping parameters might reflect the reluctance of reporting bullies, and this study aimed to extend their conclusion to both dimensions. The MM‐MH‐RM algorithm, the original MH‐RM algorithm, and the fully Bayesian estimation were used to obtain the item estimates of the two‐dimensional 4PLM.…”

“…The proposed MM-MH-RM is a natural extension of the mixture-modelling approach to the multidimensional scenario, and it typically differs from the original MH-RM (Cai, 2008) algorithm in that it introduces an extra latent variable to separate the probabilities of guessing, slipping, and ability-based responding, just as the modification of the BE3M (Zheng et al, 2021) to Bayesian EM (Mislevy, 1986) and the revision of Gibbs-within-Gibbs sampler (Culpepper, 2015) to the straightforward MCMC approach (Patz & Junker, 1999). In contrast to Zheng et al's (2021) BE3M, the proposed MM-MH-RM is specially designed for multidimensional tests; it adopts a MH sampler to approximate the high-dimensional integration rather than fixed one-dimensional Gaussian quadrature and then updates the item parameters via the RM method rather than Newton-Raphson iterations. Furthermore, unlike the time-consuming fully Bayesian MCMC methods, the MM-MH-RM only samples ability parameters but uses gradient, information, and priors of item parameters to obtain MAP estimates to guarantee stable accuracy and fast convergence.…”

“…Based on the successful application of mixture-modelling reformulation for the unidimensional 4PLM (Zheng et al, 2021), this study aims to propose a mixture-modelling-based Bayesian MH-RM (MM-MH-RM) algorithm to obtain the maximum a posteriori (MAP) estimates for the confirmatory M4PLM by combining the latent variable augmentation approach with the MH-RM algorithm. Moreover, the MM-MH-RM algorithm is expected not only to obtain fast and stable estimates but also reduce the dependence of the original MH-RM algorithm on initial values.…”

“…Another obstacle to using the 4PLM is the estimation issue. In the unidimensional scenario, researchers recently developed several methods to obtain relatively stable estimates of the 4PLM with a moderate sample size (about 5000 examinees), including, for example, the fully Bayesian Markov chain Monte Carlo (MCMC) algorithm (Culpepper, 2015), the mixture‐modelling‐based Bayesian expectation–maximization (EM) (so‐called BE3M) algorithm (Guo et al, 2020; Zheng et al, 2021), and the mixed stochastic approximation EM algorithm (Meng & Xu, 2022). However, fast and stable estimation remains a challenge for the multidimensional 4PLM (M4PLM) due to (1) the exponential growth of numerical integration quadrature (curse of dimensionality), (2) the sparse data and information for guessing and unslipping parameters, and (3) extremely high estimation time.…”